7:09 PM Sun 16 Mar - 35\% mylab.pearson.com Kurulus Osm... Can't Open P... CUNY Login The Evolutio... Do Homewor... Discord MAT 220-S25 Rasheed Larry 03/16/25 7:09 PM \( \equiv \) Homework: Partial Fractions, part 2 Question 3, 8.5.25 HW Score: 80\%, 4 of 5 points ® Points: 0 of 1 Save Question list Express the integrand as a sum of partial fractions and evaluate the integral. \[ \int \frac{24 s+24}{\left(s^{2}+1\right)(s-1)^{3}} d s \] Question 1 Question 2 \[ \int \frac{24 s+24}{\left(s^{2}+1\right)(s-1)^{3}} d s=\square \] Question 3 Question 4 Question 5 Help me solve this View an example Get more help Clear all Check answer

To solve the integral \(\int \frac{24 s+24}{\left(s^{2}+1\right)(s-1)^{3}} d s\), you'll first want to use partial fraction decomposition. You’ll break down the integrand into simpler fractions: \[ \frac{A s + B}{s^2 + 1} + \frac{C}{(s - 1)^3} + \frac{D}{(s - 1)^2} + \frac{E}{(s - 1)} \] Next, multiply through by the denominator \((s^2 + 1)(s - 1)^3\) and collect like terms to find constants A, B, C, D, and E. After you determine these, you can integrate each term separately. Don't forget, a common mistake is losing track of the signs when expanding polynomials, so double-check your work as you decompose! Once done, evaluate the resulting integrals from each component and combine the results to get your final answer for the integral. This can be a great practice in integrating rational functions, giving you some solid skills for calculus!